How do I rig Walschaerts valve gear in Blender?
I think the problem is that you have the valve stem and crosshead rigged to move together.  In an actual valve, the motion is more complex

I'm not good at drivers, but what I think you need to do is match the actual cycle more closely.

Using the animation from wiki starting with the valve stem all the way to the right end of its cylinder, the piston end of the cross head in the middle of its cylinder


Steam enters, pushing the piston to the left end of its cylinder
As the piston approaches the end of the cylinder the various mechanical linkages pull the valve stem to the left.
Once the valve stem is all the way to the right, the direction of steam reverses. This reverses the direction of the piston.
The piston moves all the way to the right
Mechanical linkages pull the valve stem to the left


end of cycle.

A driver that causes the cylinder head to move as if it were driven by steam and a set of bones that match the actual mechanical linkages, down to parenting, should cause everything to behave properly.